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Apollon suffered a blow even before a ball was kicked with the withdrawal of Hungarian midfielder Gabor Korolovsky - a pulled calf muscle means he is also a big doubt for next week's return leg in Nicosia.
Cork, too, had some late drama to deal with when John O'Flynn failed a late fitness test on a groin strain and failed to make the bench.
Cork, minus George O'Callaghan as the rift between him and the manager deepened, fielded the team which won 2-0 at Longford at the weekend.
Cork made a bright start against an Apollon side who went with a 3-5-2 formation and Denis Behan had the ball in the net after just six minutes - only to be called back for offside.
Neale Fenn and Behan again were close to giving the hosts the lead inside the opening quarter as the home side held their own against physically stronger Cypriot opposition.
As the strains of the Champions' League signature tune faded in the evening sunshine, Cork soon got to grips with their first taste of Champions' League action.
The game was in the 28th minute before Apollon forced their first corner when Jassim Arig's long range effort was deflected wide of the target.
Polish centre-forward Ukas Sosin, top scorer in the Cypriot league for the last three years, laid the ball back for midfielder Arig - but again he shot high over Michael Devine's crossbar.
The first booking of the night came on 41 minutes when Woods was yellow carded for a foul on Leandro Alvarez.
The only goal of the night came in the 62nd minute, when Woods scored with a left-footed drive into the top corner after Joe Gamble fed him the ball on the edge of the penalty area.
But Cork had a late scare when Sosin hit the post with a curling free-kick three minutes from time.